To find the value of $x$ when $f(g(x)) = 9$, follow these steps:

Given:

$f(x) = -2x + 3$ $g(x) = \frac{1}{2}x - 2$

Step 1: Find $f(g(x))$

Substitute $g(x)$ into $f(x)$: $f(g(x)) = f\left(\frac{1}{2}x - 2\right)$ Now, replace $x$ in $f(x) = -2x + 3$ with $\frac{1}{2}x - 2$: $f(g(x)) = -2\left(\frac{1}{2}x - 2\right) + 3$ Distribute the $-2$: $f(g(x)) = -2 \cdot \frac{1}{2}x + 4 + 3$ Simplify: $f(g(x)) = -x + 7$

Step 2: Set $f(g(x)) = 9$

$-x + 7 = 9$

Step 3: Solve for $x$

Subtract 7 from both sides: $-x = 2$ Multiply by $-1$ to solve for $x$: $x = -2$

So, $x = -2$.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. What is $g(f(x))$ for the given functions?
2. How would the solution change if $f(g(x)) = -9$?
3. What is the inverse of $f(x)$ and $g(x)$?
4. How do the graphs of $f(x)$ and $g(x)$ look?
5. Can you find $f(x) = 0$ for the given $f(x)$?

Tip: When solving equations involving function compositions, always carefully apply the functions step by step to avoid mistakes.